Yes Sunday was named after the sun because many groups worshipped the sun. It's common to adopt the traditions of the group and incorporate them into the new religion. Judaism's Sabbath is on Saturday because that religion adopted practices from Saturn worshippers. Christians should be celebrating Jesus birth in spring time rather than around the winter solstice.
I wouldn't say christians compromised their religion because the formation of Christianity included adopting stories and practices from preexisting religions. Aspect of Jesus were taken from Mithras and other deities in nearby regions. Noah's ark and the flood is the same (with small differences) story throughout the world. The Sumerians 5k+ years ago had almost exactly the same story.
Yes they catered to pagans because nature worship was common throughout Europe. The new religion uses traditions of the existing people in order to get the people to practice the new religion. Almost all religions do this.